# Микроэкономика Высшей Школы Экономики. Олигополия

Разберем некоторые задания из Problem set 25-26 от 2019 года

Problem set 25-26

Problem 1

Assume the market structure is best characterized by monopolistic competition. Firm 1 is one of the producers in this market. The inverse demand for this firm is 𝑝 = 500 − 9,75𝑄. Firm 1’s cost function is 𝑇𝐶(𝑄) = 0,25𝑄2 + 6.
Determine its profit maximizing level of output and the price charged to customers.

Is the market in a long-run equilibrium?

Это модель ценообразования за лидером или квазимонополия. Идея такая: фирма лидер ведет себя как монополист и максимизирует прибыль исходя из равенства MR=MC. Поскольку при монополии образуется своего рода дефицит, то остаточный спрос покрывают другие мелкие фирмы, которые являются по отношению друг к другу совершенными конкурентами. Для них выполняется равенство P=MC.

Однако нам здесь не даны функции затрат фирм-аутсайдеров, поэтому мы можем найти цену и объем выпуска фирмы из равенства MR=MC. Я считаю, что нам дана функция спроса на продукцию лидера.

Рынок не находится в долгосрочном равновесии. Фишка этого рынка в том, что при наличии положительной экономической прибыли, в отрасль приходят новые фирмы и в долгосрочном равновесии система приходит к условиям совершенной конкуренции, то есть будет выполняться равенство цены предельным затратам P=MC.

Problem 2

The industry is dominated by two major competitors: A and B. Both companies have similar technology and their costs functions are given by 𝑇𝐶(𝑦𝐴) = 20𝑦𝐴 and 𝑇𝐶(𝑦𝐵) = 20𝑦𝐵. In order to simplify our analysis, we assume there are no fixed costs. The inverse demand function for goods is estimated to be 𝑝(𝑦) = 200 − 𝑦.

(a) Find analytically the best response functions for A and B, 𝑅𝐹 (𝑦 ) and 𝑅𝐹 (𝑦 ), and plot

it in the graph.
(b) Find analytically the market price, the level of individual and aggregate production in a Cournot-Nash equilibrium. Also find the level of profit of each individual firm. Show the equilibrium in your graph.
(c) What is the DWL associated with oligopolistic trading by the two firms?
(d) Suppose the firms A and B form a cartel. What is the aggregate level of production, and profit per firm given collusion? Does collusion benefit the two producers?
(e) Find a DWL given collusion, and compare it to the one from (c). Which loss is greater, why?
(f) Is the considered cartel sustainable if the interactions, as described above, are only in the short run? Why? How about if the market interactions are repeated? Why?

Здесь мы видим, что в отрасли соревнуются 2 фирмы, то есть дуополия. Для нахождения выпуска и равновесной цены нам необходимо составить функции реакции обеих фирм. Функции реакции выводятся из условия максимизации прибыли, при этом мы учитываем, что выпуск Q=Qa+Qb

Запишем прибыль первой фирмы:

Па=ya(200-ya-yb)-20ya = 200ya-ya^2 -ya*yb-20ya = 180ya-ya^2-ya*yb->max

Достигаем максимума при: 180-2ya-yb=0 — Это уравнение реакции первой фирмы

Аналогично мы получим уравнение реакции второй фирмы:

180-2yb-ya=0 — Это уравнение реакции второй фирмы.

Согласно модели Курно, обе фирмы буду выпускать по 60 единиц продукции. Для нахождения равновесия мы решали систему из двух уравнений, то есть искали точку пересечения кривых реакции). Q = 120. P=80.

Problem 3

There are two identical firms in an industry, 1 and 2, each with cost function 𝑇𝐶𝑖 = 10𝑦𝑖,
𝑖 = 1,2. The industry demand curve is 𝑝(𝑦) = 100 − 5𝑦 where industry output, 𝑦, is the sum of the two firms’ outputs (𝑦1 + 𝑦2).
(a) If each firm makes its output decisions on the assumption that the other will not react to its choices, what is the equilibrium output for each firm? What is the equilibrium price? (b)Suppose that each firm takes it in turn to choose its level of output, on the assumption that the other’s output level is fixed. Would the process of adjustment be stable? (See Varian, 28.7 Adjustment to Equilibrium)

(c) Suppose that firm 1 introduces a cost-saving innovation, so that its cost curve becomes 𝑇𝐶1 = 8𝑦1. Firm 2’s cost curve and the industry demand curve are unchanged. What happens to the equilibrium quantity produced by each firm and to market price?
(d) Suppose these two firms are now playing a Stackelberg game, with firm 1 as leader and firm 2 as follower. What are the equilibrium levels of output and industry price? How does your answer change if firm 1 has the cost function given in (c)?

(e) Show that in equilibrium firm 1’s isoprofit curve is tangent to firm 2’s reaction function.

Problem 4

Consider that there are 2 firms selling the goods for some market. Given their cost functions: 𝑐 (𝑞 ) = 𝑞2 + 7𝑞 and 𝑐 (𝑞 ) = 𝑞2 + 13𝑞 . The demand for their production is

11112222 given as well: 𝐷(𝑝) = 100 − 𝑝.

(a) Assume that the firms compete by Cournot. Calculate equilibrium prices, outputs and profits.
(b) Sketch the equilibrium on 2 graphs: in terms of output, price and profits.

𝐴𝐵 𝐵𝐴

(c) Now assume that the game is still asymmetric and the cost functions are given as
𝑐(𝑞𝑖) = 𝑐𝑖𝑞𝑖, 𝑎 > 𝑐1 > 𝑐2 > 0. The demand function is 𝐷(𝑝) = 𝑎 − 𝑝, 𝑎 > 0. Compute the Cournot equilibrium (output, price and profits).
(d)Now let the game be symmetric, but the number of firms is now 𝑐(𝑞𝑖) = 𝑐𝑞𝑖, 𝑖 = 1…𝑛. The demand is still the same 𝐷(𝑝) = 𝑎 − 𝑝, 𝑎 > 0. Compute the Cournot equilibrium (output, price and profits).
(e) What is the price if 𝑛 → ∞?
(f) (optional*) Calculate the equilibrium (output, price and profits) for an asymmetric Cournot oligopoly with 𝑛 firms on the market. 𝑐(𝑞𝑖) = 𝑐𝑖𝑞𝑖, 𝑎 > 𝑐1 > 𝑐2 > ⋯ > 𝑐𝑛 > 0, 𝐷(𝑝) = 𝑎 − 𝑝, 𝑎 > 0. How the asymmetry affects the price?

Problem 5

Two firms compete Stackelberg oligopoly and their cost functions are 𝑇𝐶𝐴 = 𝑞𝐴, 𝑇𝐶𝐵 = 𝑐𝑞𝐵, 𝑐 > 1. The firm 𝐴 acts as a leader and 𝐵 accept it. The demand function is 𝐷(𝑝) = 𝑎 − 𝑝, 𝑎>𝑐.
(a) Calculate Stackelberg equilibrium (output, price and profits).
(b) How the equilibrium prices and outputs will change if the marginal costs of the firm 𝐵 increase?
(c) What firm will response higher change in output on the increasing marginal costs of 𝐵? (d) Now, let 𝑐 = 1 and the firm 𝐵 decides to reject 𝐴 leadership and fight (𝐵 decides to act like a leader too), however 𝐴 still acts like a leader. What are the prices, outputs and profits now?
(e) Now assume that the firm 𝐵 decides to reject 𝐴 leadership and fight (𝐵 plays Cournot), however 𝐴 still acts like a leader. What are the prices, outputs and profits now?
(f) Compare the equilibrium in (d and e) with the equilibrium that would take place if 𝐴 and 𝐵 stop their leadership behavior
(g) What equilibrium (output, price and profits) would be established if both firms agree to collaborate and form a cartel?
(h) Why Stackelberg oligopoly can appear?

Problem 6

We assume an oligopoly market with two firms. The indirect demand functions for the firms’ goods are 𝑝1 = 𝛼 − 𝑞1 − 𝛾𝑞2 and 𝑝2 = 𝛼 − 𝑞2 − 𝛾𝑞1, where 𝛼 > 0 and −1 < 𝛾 < 1. The firms have the same cost function, which is given by 𝑐(𝑞𝑖) = 𝑐𝑞𝑖, where 0 ≤ 𝑐 < 𝛼.
(a) Calculate the Cournot-Nash equilibrium. What is the market price for each good in this equilibrium? Are 𝑞1 and 𝑞2 strategic substitutes or strategic complements?
(b) Invert the two indirect demand functions so that you get two direct demand functions. (c) Calculate the Bertrand-Nash equilibrium. What is the market price for each good in this equilibrium? Are 𝑝1 and 𝑝2 strategic substitutes or strategic complements?
(d) Which model (quantity setting or price setting) gives rise to the lowest market price?

Problem 7

Suppose there are firm 1 and firm 2 in a differentiated goods market. The demand functions are 𝑦1 = 34 − 𝑝1 + 1 𝑝2 and 𝑦2 = 40 − 𝑝2 + 1 𝑝1. The firms’ cost functions are

32
𝑐1(𝑦1) = 24𝑦1 and 𝑐2(𝑦2) = 20𝑦2. They compete with each other through their choices of

price.
(a) Calculate the equilibrium in which firm 1 is the price leader in this market, and firm 2 is the price follower. Indicate prices, output levels, and individual profits.
(b) How do prices, output levels, and individual profits change if firm 2 is the price leader in this market, and firm 1 is the price follower?

Problem 8

Suppose 80 small firms with identical cost functions 𝑇𝐶 = 2 + 8𝑞2 operate in the industry. 𝑖𝑖

Also there is a dominant firm with the cost function 𝑇𝐶 = 20 + 0,275𝑞2.

Industry demand is 𝑄𝐷 = 256– 3𝑝.
What is the market price?
How will the industry output be divided between the leader and followers?